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If we have an $n \times n$ positive semidefinite matrix $A$ and we have two decompositions such that $A = B B^T = C C^T$ for some $n \times n$ matrices $B$ and $C$.

Is it true that $B$ and $C$ are related by a unitary matrix? Specifically, is $B U = C$ for some unitary matrix $U$?

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  • $\begingroup$ I guess no. Try to find a counterexample by choosing $A=\pmatrix{1&0\\0&0}$. $\endgroup$ – Berci Jan 23 at 9:06

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